Mathematical Methods for Physics and Engineering : A Comprehensive Guide

23.3 Operator notation and the existence of solutions

In fact, we shall be concerned with various special cases of (23.4), which are

known by particular names. Firstly, ifg(x) = 0 then the unknown functiony(x)

appears only under the integral sign, and (23.4) is called a linear integral equation

of the first kind. Alternatively, ifg(x) = 1, so thaty(x) appears twice, once inside

the integral and once outside, then (23.4) is called a linear integral equationof

the second kind. In either case, iff(x) = 0 the equation is calledhomogeneous,

otherwiseinhomogeneous.

We can distinguish further between different types of integral equation by the

form of the integration limitsaandb. If these limits are fixed constants then the

equation is called aFredholmequation. If, however, the upper limitb=x(i.e. it

is variable) then the equation is called aVolterraequation; such an equation is

analogous to one with fixed limits but for which the kernelK(x, z)=0forz>x.

Finally, we note that any equation for which either (or both) of the integration

limits is infinite, or for whichK(x, z) becomes infinite in the range of integration,

is called asingularintegral equation.

23.3 Operator notation and the existence of solutions

There is a close correspondence between linear integral equations and the matrix

equations discussed in chapter 8. However, the former involve linear, integral rela-

tions between functions in an infinite-dimensional function space (see chapter 17),

whereas the latter specify linear relations among vectors in a finite-dimensional

vector space.

Since we are restricting our attention to linear integral equations, it will be

convenient to introduce the linear integral operatorK,whoseactiononan

arbitrary functionyis given by

Ky=

∫b

a

K(x, z)y(z)dz. (23.5)

This is analogous to the introduction in chapters 16 and 17 of the notationLto

describe a linear differential operator. Furthermore, we may define the Hermitian

conjugateK†by

K†y=

∫b

a

K∗(z, x)y(z)dz,

where the asterisk denotes complex conjugation and we have reversed the order

of the arguments in the kernel.

It is clear from (23.5) thatKis indeed linear. Moreover, sinceKoperates on

the infinite-dimensional space of (reasonable) functions, we may make an obvious

analogy with matrix equations and consider the action ofKon a functionfas

that of a matrix on a column vector (both of infinite dimension).

When written in operator form, the integral equations discussed in the pre-

vious section resemble equations familiar from linear algebra. For example, the